Flux — Question 10

Question 10

Three independent flux measurements reconstruct a uniform flow

Three small planar windows sample the same uniform velocity. Each has area $.100$ m$^2$. Their normals are $\hat{\mathbf x}$, $\hat{\mathbf y}$ and $(\hat{\mathbf x}+\hat{\mathbf y}+\hat{\mathbf z})/\sqrt3$. Signed fluxes are $.200$, $-.100$ and $.300/\sqrt3$ m$^3$/s, respectively.

1. Reconstruct the three Cartesian velocity components and its speed.

2. Predict the flux through a fourth equal-area window with normal $(\hat{\mathbf x}-\hat{\mathbf z})/\sqrt2$. Explain whether its zero reading would imply zero speed.

Question 10 - Solution

Three independent flux measurements reconstruct a uniform flow

(a) Method and calculation

$$v_x=Q_1/A=2.00,\quad v_y=Q_2/A=-1.00\qquad(\mathrm{m/s}).$$$$v_x+v_y+v_z=\sqrt3Q_3/A=3.00\Rightarrow v_z=2.00\,\mathrm{m/s}.$$

(b) Interpretation and further calculation

$$v=\sqrt{2^2+(-1)^2+2^2}=3.00\,\mathrm{m/s}.$$$$Q_4=\frac A{\sqrt2}(v_x-v_z)=0.$$The fourth normal is perpendicular to the reconstructed velocity; its zero reading is compatible with a moving fluid.

Physical check

Reconstruction requires independent normals. Repeating parallel measurements cannot recover missing transverse components.